In short: Trigonometry spans four years of school maths - from basic ratios in Class 9-10 to compound angles in Class 11, and inverse trig functions with calculus in Class 12. This single reference sheet covers every formula you will need across CBSE boards, JEE, and NEET. Bookmark it, print it, or keep it on your phone for instant revision.
Why One Trigonometry Sheet Covers Four Years
Trigonometry is not four separate topics taught across four classes. It is one continuous subject that builds layer by layer. Class 9-10 gives you the foundation - basic ratios and identities. Class 11 expands into compound angles, multiple angles, and transformations. Class 12 applies everything to calculus - derivatives and integrals of trig functions.
The problem students face is fragmentation. Your Class 10 notes are in one notebook, Class 11 in another, and Class 12 in a third. When a JEE problem requires a Class 10 identity combined with a Class 12 integration technique, you waste time hunting for the right formula.
This sheet solves that. Every trig formula from Class 9 through 12, organised by category. One page, four years of trigonometry.
Foundation - Basic Trigonometric Ratios (Class 9-10)
These six ratios are where everything begins. Every single trig formula in Class 11 and 12 is built on top of these.
| Ratio | Definition | Reciprocal |
|---|---|---|
| sin(A) | Opposite / Hypotenuse | cosec(A) = 1/sin(A) |
| cos(A) | Adjacent / Hypotenuse | sec(A) = 1/cos(A) |
| tan(A) | Opposite / Adjacent = sin(A)/cos(A) | cot(A) = 1/tan(A) = cos(A)/sin(A) |
Reciprocal relationships:
- sin(A) x cosec(A) = 1
- cos(A) x sec(A) = 1
- tan(A) x cot(A) = 1
Standard Angle Values (Class 9-10)
This table is worth memorising completely. You will use these values hundreds of times.
| Angle | 0 deg | 30 deg | 45 deg | 60 deg | 90 deg |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/sqrt(2) | sqrt(3)/2 | 1 |
| cos | 1 | sqrt(3)/2 | 1/sqrt(2) | 1/2 | 0 |
| tan | 0 | 1/sqrt(3) | 1 | sqrt(3) | undefined |
| cosec | undefined | 2 | sqrt(2) | 2/sqrt(3) | 1 |
| sec | 1 | 2/sqrt(3) | sqrt(2) | 2 | undefined |
| cot | undefined | sqrt(3) | 1 | 1/sqrt(3) | 0 |
Memory trick: For sin values from 0 to 90 degrees, use sqrt(0)/2, sqrt(1)/2, sqrt(2)/2, sqrt(3)/2, sqrt(4)/2. That gives you 0, 1/2, 1/sqrt(2), sqrt(3)/2, 1. Cos values are the same sequence in reverse order.
Fundamental Pythagorean Identities (Class 9-10)
These three identities are the backbone of trigonometry. Every identity simplification problem in board exams traces back to one of these.
| Identity | Alternate Form 1 | Alternate Form 2 |
|---|---|---|
| sin^2(A) + cos^2(A) = 1 | sin^2(A) = 1 - cos^2(A) | cos^2(A) = 1 - sin^2(A) |
| 1 + tan^2(A) = sec^2(A) | tan^2(A) = sec^2(A) - 1 | sec^2(A) - tan^2(A) = 1 |
| 1 + cot^2(A) = cosec^2(A) | cot^2(A) = cosec^2(A) - 1 | cosec^2(A) - cot^2(A) = 1 |
Exam tip: When you see sec^2(A) or cosec^2(A) in a problem, immediately think of replacing them using these identities. This is the single most common simplification step in Class 10 board problems.
Complementary Angle Formulas (Class 9-10)
| Formula | Example |
|---|---|
| sin(90 - A) = cos(A) | sin(60) = cos(30) = sqrt(3)/2 |
| cos(90 - A) = sin(A) | cos(60) = sin(30) = 1/2 |
| tan(90 - A) = cot(A) | tan(60) = cot(30) = sqrt(3) |
| cot(90 - A) = tan(A) | cot(60) = tan(30) = 1/sqrt(3) |
| sec(90 - A) = cosec(A) | sec(60) = cosec(30) = 2 |
| cosec(90 - A) = sec(A) | cosec(60) = sec(30) = 2/sqrt(3) |
Sign Convention by Quadrant (Class 11)
| Quadrant | Angle Range | sin | cos | tan |
|---|---|---|---|---|
| I | 0 to 90 deg | + | + | + |
| II | 90 to 180 deg | + | - | - |
| III | 180 to 270 deg | - | - | + |
| IV | 270 to 360 deg | - | + | - |
Memory aid: "All Students Take Coffee" - All positive in Q1, Sin positive in Q2, Tan positive in Q3, Cos positive in Q4.
Allied Angle Formulas (Class 11)
These formulas let you find values of trig functions for any angle beyond 0-90 degrees.
Angles involving (180 - A):
- sin(180 - A) = sin(A)
- cos(180 - A) = -cos(A)
- tan(180 - A) = -tan(A)
Angles involving (180 + A):
- sin(180 + A) = -sin(A)
- cos(180 + A) = -cos(A)
- tan(180 + A) = tan(A)
Angles involving (360 - A):
- sin(360 - A) = -sin(A)
- cos(360 - A) = cos(A)
- tan(360 - A) = -tan(A)
Angles involving (-A), or negative angles:
- sin(-A) = -sin(A)
- cos(-A) = cos(A)
- tan(-A) = -tan(A)
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Compound Angle Formulas (Class 11)
These are the workhorses of Class 11 trigonometry. JEE and NEET questions use them constantly.
| Formula | Expression |
|---|---|
| sin(A + B) | sin(A)cos(B) + cos(A)sin(B) |
| sin(A - B) | sin(A)cos(B) - cos(A)sin(B) |
| cos(A + B) | cos(A)cos(B) - sin(A)sin(B) |
| cos(A - B) | cos(A)cos(B) + sin(A)sin(B) |
| tan(A + B) | (tan A + tan B) / (1 - tan A tan B) |
| tan(A - B) | (tan A - tan B) / (1 + tan A tan B) |
| cot(A + B) | (cot A cot B - 1) / (cot B + cot A) |
| cot(A - B) | (cot A cot B + 1) / (cot B - cot A) |
Double Angle Formulas (Class 11)
Put A = B in the compound angle formulas and you get double angle formulas.
| Formula | Expression |
|---|---|
| sin(2A) | 2 sin(A) cos(A) |
| cos(2A) - Form 1 | cos^2(A) - sin^2(A) |
| cos(2A) - Form 2 | 2 cos^2(A) - 1 |
| cos(2A) - Form 3 | 1 - 2 sin^2(A) |
| tan(2A) | 2 tan(A) / (1 - tan^2(A)) |
Derived half-angle results:
- sin^2(A) = (1 - cos 2A) / 2
- cos^2(A) = (1 + cos 2A) / 2
Triple Angle Formulas (Class 11)
| Formula | Expression |
|---|---|
| sin(3A) | 3 sin(A) - 4 sin^3(A) |
| cos(3A) | 4 cos^3(A) - 3 cos(A) |
| tan(3A) | (3 tan A - tan^3 A) / (1 - 3 tan^2 A) |
Sum-to-Product Formulas (Class 11)
These convert sums of trig functions into products - useful for solving equations.
| Formula | Expression |
|---|---|
| sin C + sin D | 2 sin((C+D)/2) cos((C-D)/2) |
| sin C - sin D | 2 cos((C+D)/2) sin((C-D)/2) |
| cos C + cos D | 2 cos((C+D)/2) cos((C-D)/2) |
| cos C - cos D | -2 sin((C+D)/2) sin((C-D)/2) |
Product-to-Sum Formulas (Class 11)
The reverse of sum-to-product. Used frequently in integration (Class 12).
| Formula | Expression |
|---|---|
| 2 sin A cos B | sin(A+B) + sin(A-B) |
| 2 cos A sin B | sin(A+B) - sin(A-B) |
| 2 cos A cos B | cos(A-B) + cos(A+B) |
| 2 sin A sin B | cos(A-B) - cos(A+B) |
General Solutions of Trig Equations (Class 11)
| Equation | General Solution |
|---|---|
| sin(x) = 0 | x = n pi, where n is an integer |
| cos(x) = 0 | x = (2n+1) pi/2 |
| tan(x) = 0 | x = n pi |
| sin(x) = sin(alpha) | x = n pi + (-1)^n alpha |
| cos(x) = cos(alpha) | x = 2n pi +/- alpha |
| tan(x) = tan(alpha) | x = n pi + alpha |
| sin(x) = 1 | x = (4n+1) pi/2 |
| cos(x) = 1 | x = 2n pi |
Inverse Trigonometric Functions (Class 12)
Domain and Range
| Function | Domain | Range (Principal Value) |
|---|---|---|
| sin^(-1)(x) | [-1, 1] | [-pi/2, pi/2] |
| cos^(-1)(x) | [-1, 1] | [0, pi] |
| tan^(-1)(x) | (-infinity, infinity) | (-pi/2, pi/2) |
| cot^(-1)(x) | (-infinity, infinity) | (0, pi) |
| sec^(-1)(x) | (-infinity, -1] U [1, infinity) | [0, pi] - {pi/2} |
| cosec^(-1)(x) | (-infinity, -1] U [1, infinity) | [-pi/2, pi/2] - {0} |
Key Inverse Trig Properties
| Property | Condition |
|---|---|
| sin^(-1)(x) + cos^(-1)(x) = pi/2 | for x in [-1, 1] |
| tan^(-1)(x) + cot^(-1)(x) = pi/2 | for all x |
| sec^(-1)(x) + cosec^(-1)(x) = pi/2 | for |
| sin^(-1)(-x) = -sin^(-1)(x) | for x in [-1, 1] |
| cos^(-1)(-x) = pi - cos^(-1)(x) | for x in [-1, 1] |
| tan^(-1)(-x) = -tan^(-1)(x) | for all x |
| tan^(-1)(x) + tan^(-1)(y) = tan^(-1)((x+y)/(1-xy)) | when xy < 1 |
| tan^(-1)(x) - tan^(-1)(y) = tan^(-1)((x-y)/(1+xy)) | when xy > -1 |
| 2 tan^(-1)(x) = sin^(-1)(2x/(1+x^2)) | for |
| 2 tan^(-1)(x) = cos^(-1)((1-x^2)/(1+x^2)) | for x >= 0 |
| 2 tan^(-1)(x) = tan^(-1)(2x/(1-x^2)) | for |
Derivatives of Trigonometric Functions (Class 12)
| Function | Derivative |
|---|---|
| d/dx [sin(x)] | cos(x) |
| d/dx [cos(x)] | -sin(x) |
| d/dx [tan(x)] | sec^2(x) |
| d/dx [cot(x)] | -cosec^2(x) |
| d/dx [sec(x)] | sec(x) tan(x) |
| d/dx [cosec(x)] | -cosec(x) cot(x) |
Derivatives of Inverse Trig Functions
| Function | Derivative |
|---|---|
| d/dx [sin^(-1)(x)] | 1 / sqrt(1 - x^2) |
| d/dx [cos^(-1)(x)] | -1 / sqrt(1 - x^2) |
| d/dx [tan^(-1)(x)] | 1 / (1 + x^2) |
| d/dx [cot^(-1)(x)] | -1 / (1 + x^2) |
| d/dx [sec^(-1)(x)] | 1 / ( |
| d/dx [cosec^(-1)(x)] | -1 / ( |
Integrals of Trigonometric Functions (Class 12)
Basic Trig Integrals
| Integral | Result |
|---|---|
| integral of sin(x) dx | -cos(x) + C |
| integral of cos(x) dx | sin(x) + C |
| integral of tan(x) dx | -ln |
| integral of cot(x) dx | ln |
| integral of sec(x) dx | ln |
| integral of cosec(x) dx | ln |
| integral of sec^2(x) dx | tan(x) + C |
| integral of cosec^2(x) dx | -cot(x) + C |
| integral of sec(x) tan(x) dx | sec(x) + C |
| integral of cosec(x) cot(x) dx | -cosec(x) + C |
Integrals Leading to Inverse Trig Functions
| Integral | Result |
|---|---|
| integral of dx / sqrt(1 - x^2) | sin^(-1)(x) + C |
| integral of dx / (1 + x^2) | tan^(-1)(x) + C |
| integral of dx / (x sqrt(x^2 - 1)) | sec^(-1)(x) + C |
| integral of dx / sqrt(a^2 - x^2) | sin^(-1)(x/a) + C |
| integral of dx / (a^2 + x^2) | (1/a) tan^(-1)(x/a) + C |
| integral of dx / (x sqrt(x^2 - a^2)) | (1/a) sec^(-1)(x/a) + C |
How to Study Trigonometry Across Four Years
Class 9-10 - Build the Foundation
At this stage, focus on three things: memorise the standard angle table, understand the three Pythagorean identities, and practise proving simple identities. Every identity proof in Class 10 boards uses the same strategy - convert everything to sin and cos, then simplify using sin^2 + cos^2 = 1.
Do not skip the complementary angle formulas. Questions like "sin(90 - A) = cos(A), verify for A = 30 degrees" appear in every board exam.
Class 11 - Master the Expansions
Compound angle formulas are where most students first struggle. The trick is to not memorise all four separately. Learn sin(A+B) and cos(A+B), then derive sin(A-B) by replacing B with -B, and cos(A-B) similarly.
For sum-to-product and product-to-sum, practise converting both ways. JEE problems often require you to go from a product to a sum (for integration) or from a sum to a product (for solving equations).
Class 12 - Apply to Calculus
Inverse trig functions have strict domain and range restrictions. If you ignore these, you will get wrong answers in board exams. Always check whether your input value falls within the domain before applying a formula.
For integration, the key insight is recognising patterns. When you see 1/(1 + x^2), think tan^(-1). When you see 1/sqrt(1 - x^2), think sin^(-1). These pattern recognition skills come only from solving 30-40 problems.
Five Common Trigonometry Mistakes
1. Confusing sin^(-1)(x) with 1/sin(x). The notation sin^(-1)(x) means the inverse sine function (arcsin), not the reciprocal. The reciprocal of sin(x) is cosec(x). This confusion costs marks in Class 12 board exams.
2. Applying compound angle formulas with wrong signs. In cos(A+B), the middle sign is minus: cos(A)cos(B) - sin(A)sin(B). In cos(A-B), it is plus. This is opposite to what you might expect. Double-check every time.
3. Forgetting general solution conditions. For tan^(-1)(x) + tan^(-1)(y) = tan^(-1)((x+y)/(1-xy)), the condition xy < 1 is essential. If xy > 1, you need to add or subtract pi. Board examiners specifically test this.
4. Not converting degrees to radians. In Class 12, all calculus-based trig problems use radians. If your calculator is in degree mode while differentiating sin(x), every answer is wrong.
5. Skipping the domain check for inverse trig. sin^(-1)(2) does not exist because the domain of sin^(-1) is [-1, 1]. Students sometimes substitute values blindly without checking domain restrictions.
FAQ
How many trigonometry formulas do I need for board exams?
For Class 10 boards, roughly 15-20 formulas (basic ratios, identities, complementary angles, standard values). For Class 12 boards, add another 30-40 (compound angles, inverse trig, derivatives, integrals). For JEE, you need practically all of them - about 60-70 total.
What is the fastest way to memorise the standard angle table?
Use the sqrt pattern: sin(0) = sqrt(0)/2, sin(30) = sqrt(1)/2, sin(45) = sqrt(2)/2, sin(60) = sqrt(3)/2, sin(90) = sqrt(4)/2. Cos values are the reverse. Once you know sin and cos, derive tan = sin/cos.
Are triple angle formulas important for CBSE?
They occasionally appear in Class 11 exams and JEE. They are not as frequently tested as compound and double angle formulas, but knowing sin(3A) = 3sinA - 4sin^3A is useful for solving cubic trig equations.
How do inverse trig functions connect to the basic ratios?
If sin(30 deg) = 1/2, then sin^(-1)(1/2) = 30 deg = pi/6 radians. The inverse function answers the question "what angle gives this ratio?" The principal value branch ensures each input maps to exactly one output.
Which trigonometry topics carry the most marks in Class 12?
Inverse trigonometric functions typically carry 4-6 marks in CBSE Class 12 boards. Integration problems involving trig functions can carry another 6-8 marks. Together, trig-based questions often account for 12-15 marks of the total paper.
Can I use this sheet for JEE and NEET preparation?
Yes. This sheet covers the complete NCERT trigonometry syllabus across all four years, which is the base syllabus for both JEE and NEET. JEE Mains and Advanced may require faster application of these formulas, but the formulas themselves are the same.
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